## Abstract

For a graph G with vertex set {v1, . . ., v
_{n}
}, let P(G) be an n × n matrix whose (i, j)-entry is the maximum number of internally disjoint viv
_{j}
-paths in G, if i ≠ j, and zero otherwise. The sum of absolute values of the eigenvalues of P(G) is called the path energy of G, denoted by PE. We prove that PE of a connected graph G of order n is at least 2(n− 1) and equality holds if and only if G is a tree. Also, we determine PE of a unicyclic graph of order n and girth k, showing that for every n, PE is an increasing function of k. Therefore, among unicyclic graphs of order n, the maximum and minimum PE-values are for k = n and k = 3, respectively. These results give affirmative answers to some conjectures proposed in MATCH.

Original language | English |
---|---|

Pages (from-to) | 465-470 |

Number of pages | 6 |

Journal | Match |

Volume | 81 |

Issue number | 2 |

Publication status | Published - 1 Jan 2019 |

## OECD FOS+WOS

- 1.04 CHEMICAL SCIENCES
- 1.02 COMPUTER AND INFORMATION SCIENCES
- 1.01 MATHEMATICS